Capital Budgeting: formula sheet

Key formulas

Initial cash outflow

Initial outlay = Cost of asset + Installation and transport + Additional working capital − Sale proceeds of old asset (± tax effect on its sale)

For replacement decisions, reduce the outlay by the net cash from selling the old asset.

Incremental operating cash flow (after tax)

CFAT = (Incremental sales − Incremental cash costs − Depreciation) × (1 − t) + Depreciation

Equivalent: (Sales − Cash costs) × (1 − t) + Depreciation × t. Depreciation is added back because it is non-cash.

Terminal cash flow

Terminal flow = Salvage value (after tax) + Recovery of working capital

Received in the final year, along with that year's operating cash flow.

Relevance test

Relevant if: future + cash + incremental

Sunk costs, non-cash items and unchanged allocated costs fail this test.

Initial outlay (new project)

Initial outlay = Cost of asset + Installation and transport + Increase in net working capital + Other upfront cash costs

Occurs at time 0. Pre-operating costs that are cash costs are included. Do not include interest.

Initial outlay (replacement)

Initial outlay = Cost of new asset + Increase in NWC − After-tax proceeds of old asset

After-tax proceeds = Sale price − tax on gain, or Sale price + tax saved on loss.

Tax on sale of an asset

Tax effect = t × (Sale price − Book value)

Positive means tax paid on a gain; negative means tax saved on a loss. Apply only if the question states the loss or gain is taxable or adjustable.

Operating cash flow (build-up)

OCF = (Sales − Cash costs − Depreciation) × (1 − t) + Depreciation

Equals PAT + Depreciation when there is no interest in the calculation.

Operating cash flow (shortcut)

OCF = (Sales − Cash costs) × (1 − t) + t × Depreciation

The second term is the depreciation tax shield. Useful for quick checks.

Incremental OCF (replacement)

ΔOCF = (ΔSales − ΔCash costs) × (1 − t) + t × ΔDepreciation

ΔDepreciation = depreciation on new asset − depreciation on old asset.

Terminal cash flow

Terminal cash flow = Salvage value − Tax on salvage + Recovery of NWC

Tax on salvage = t × (Salvage − Book value at end). Added to the last year's OCF.

Straight-line depreciation

Annual depreciation = (Cost − Salvage value) ÷ Life

Use the method the question gives. Depreciable cost includes installation.

Payback period (even inflows)

Payback = Initial investment ÷ Annual cash inflow

Use net cash inflow after tax, before depreciation is deducted (add back depreciation to profit after tax).

Payback period (uneven inflows)

Payback = Years fully recovered + (Unrecovered amount at start of the year ÷ Cash inflow of that year)

Assumes inflows arise evenly within the year. Multiply the fraction by 12 to get months.

Payback reciprocal

Payback reciprocal = (Annual cash inflow ÷ Initial investment) × 100 = (1 ÷ Payback) × 100

Approximates IRR only if life is at least twice the payback and inflows are constant.

Present value of an inflow

PV = Cash inflow × 1 ÷ (1 + r)^n

Use the cost of capital as r. Use the given PV factors in the exam.

Discounted payback period

Years fully recovered (on cumulative PV) + (Unrecovered PV ÷ PV of inflow of that year)

Same interpolation as simple payback, but on discounted cash flows.

Decision rule

Accept if payback ≤ cut-off period; among alternatives choose the shortest

Cut-off is set by management, not by a formula.

Average annual profit

Average annual profit = Σ (annual profit after depreciation and tax) ÷ number of years

Annual profit = cash inflow before depreciation − depreciation (− tax, if given).

Depreciation (straight line)

Annual depreciation = (Cost − Scrap value) ÷ Life in years

Use it to convert cash inflow into accounting profit.

ARR on initial investment

ARR = Average annual profit ÷ Initial investment × 100

Use when the question says original or initial investment.

Average investment

Average investment = (Initial investment + Scrap value) ÷ 2

If working capital is recovered at the end, treat it like scrap value or add it as the question directs.

ARR on average investment

ARR = Average annual profit ÷ Average investment × 100

The most commonly examined form.

Decision rule

Accept if ARR ≥ required rate; among projects, prefer the higher ARR

Rejects projects below the cut-off.

Net Present Value

NPV = Σ [Ct ÷ (1 + k)^t] − C0, for t = 1 to n

Ct is the net cash inflow in year t, k is the discount rate, C0 is the initial outlay at time 0. Add the PV of salvage value and released working capital in the final year.

NPV with scrap and working capital

NPV = PV of operating inflows + PV of salvage value + PV of working capital recovered − (initial outlay + working capital invested)

Working capital is invested at the start and recovered at the end of the project life.

Profitability Index

PI = PV of cash inflows ÷ PV of cash outflows (initial outlay)

Also called benefit-cost ratio or desirability factor.

Net Profitability Index

Net PI = NPV ÷ Initial outlay = PI − 1

Use this if the question asks for net PI.

Decision rules

Accept if NPV > 0 (PI > 1); reject if NPV < 0 (PI < 1)

For mutually exclusive projects, choose the highest NPV. Under capital rationing, rank by PI, subject to the budget and indivisibility.

IRR definition

Σ [Cash inflow(t) ÷ (1 + IRR)^t] − Initial outlay = 0

IRR is the rate at which NPV = 0.

Payback-style factor (even inflows)

Annuity factor = Initial outlay ÷ Annual cash inflow

Look up this factor in the annuity table along the row for the project's life. The rate it falls at is the IRR.

IRR by interpolation

IRR = LR + [NPV at LR ÷ (NPV at LR − NPV at HR)] × (HR − LR)

LR is the lower rate with a positive NPV. HR is the higher rate with a negative NPV. Use the sign of NPV carefully: the denominator is the sum of the two NPVs ignoring signs.

Terminal value for MIRR

TV = Σ [Cash inflow(t) × (1 + r)^(n − t)]

r is the reinvestment rate, usually the cost of capital. n is the project life. The last year's inflow is not compounded.

Modified IRR

MIRR = (TV ÷ PV of outflows)^(1/n) − 1

Use the present value of outflows at the cost of capital. If the only outflow is at time 0, it is the initial outlay.

Decision rule

Accept if IRR (or MIRR) > cost of capital

Reject if lower. For mutually exclusive projects, compare with NPV before finalising.

Net present value

NPV = Σ [Cash inflow(t) ÷ (1 + k)^t] − Initial outlay

k is the cost of capital. In a conflict, the higher NPV project is preferred for mutually exclusive projects.

Incremental IRR rule

Choose larger project if IRR of (Larger − Smaller) cash flows > k

Gives the same decision as NPV. Compute incremental flows year by year.

Equivalent annual annuity (EAA)

EAA = NPV ÷ PVAF(k, n)

PVAF is the present value annuity factor for the project's own life n. Higher EAA is better.

Equivalent annual cost (EAC)

EAC = PV of all costs ÷ PVAF(k, n)

Use for cost-only machines. Lower EAC is better.

Profitability index

PI = PV of cash inflows ÷ Initial outlay

Used to rank projects under capital rationing when projects are divisible.

Replacement chain

Common life = LCM of project lives

Repeat each project over the common life, then compare total NPVs.

Risk-adjusted discount rate

RADR = Risk-free rate + Risk premium

Use RADR to discount the expected cash flows. Higher risk means a higher premium. Under CAPM, RADR = Rf + β × (Rm − Rf).

NPV using RADR

NPV = Σ [CFt ÷ (1 + RADR)^t] − Initial outflow

Cash flows are the expected, unadjusted ones.

Certainty equivalent coefficient

α = Certain cash flow ÷ Risky cash flow

α lies between 0 and 1. A lower α means higher risk.

NPV using certainty equivalent

NPV = Σ [αt × CFt ÷ (1 + Rf)^t] − Initial outflow

Discount at the risk-free rate only. Adjust the outflow too if it is uncertain.

Sensitivity of NPV

% change in NPV ÷ % change in the variable

The variable with the largest ratio is the most critical. Use it when the base NPV is positive. If the base NPV is negative, compare the absolute change in NPV instead.

Expected NPV from scenarios

Expected NPV = Σ (Probability of scenario × NPV of scenario)

Probabilities must add up to 1.

Quick revision

• NPV = present value of cash inflows − initial outlay; accept if NPV is greater than zero.
• Profitability Index = present value of inflows ÷ initial outlay; accept if PI is greater than 1.
• IRR is the discount rate at which NPV equals zero; accept if IRR is higher than the cost of capital.
• Payback period is the time to recover the initial outlay; it ignores cash flows after payback.
• Discounted payback uses present values of cash flows, so it is longer than simple payback.
• ARR = average accounting profit ÷ investment (initial or average, as the question states); it uses profit, not cash flow.
• Depreciation is not a cash flow; include only its tax shield = depreciation × tax rate.
• Include working capital as an outflow at the start and a recovery at the end; ignore sunk costs.
• Interest on financing is not deducted from project cash flows; the discount rate covers it.
• When NPV and IRR conflict for mutually exclusive projects, prefer the higher NPV.
• MIRR assumes inflows are reinvested at the cost of capital and gives a single rate.
• For unequal lives, compare using equivalent annual annuity or a common replacement chain.

Common mistakes

• Including sunk costs such as a feasibility study already paid for. Fix: Ask whether the amount changes if you accept or reject the project. If not, exclude it.
• Deducting interest on the loan from project cash flows. Fix: Leave out financing costs; the discount rate (cost of capital) already reflects them.
• Treating depreciation as a cash outflow, or adding it back after not deducting it. Fix: Choose one method. Build-up: deduct depreciation for tax, then add it back in full. Shortcut: never deduct it, only add t × Depreciation.
• Including interest on the loan in the cash flows. Fix: Financing costs are captured in the discount rate. Leave interest out of project cash flows unless the question clearly asks otherwise.
• Using profit after tax instead of cash inflow Fix: Always add back depreciation (and other non-cash charges) to profit after tax before computing payback.
• Deducting depreciation again from investment or cash flows Fix: Depreciation is non-cash. It only affects tax. Include its tax shield through profit after tax, then add it back.
• Using cash inflow as profit without deducting depreciation. Fix: Always ask: is this figure before or after depreciation? Deduct depreciation for ARR.
• Taking average investment as half of cost, ignoring scrap value. Fix: Use (Initial investment + Scrap value) ÷ 2. With zero scrap it becomes half of cost.
• Discounting the initial outlay Fix: The outlay at time 0 has a factor of 1. Never discount it.
• Forgetting working capital recovery Fix: Show it as an outflow at year 0 and an inflow in the last year, and discount the inflow.

Exam tips

• Write a one-line definition first; it earns marks even if the rest is brief.
• In cash flow questions, show an explicit list of items excluded with reasons. Examiners give marks for this.
• Look for hidden traps: sunk costs, allocated overheads, interest and working capital recovery.
• In MCQs, remember that there is no negative marking, so attempt every question. Watch words such as 'incremental', 'sunk' and 'mutually exclusive'.
• Use a year-wise table with Year 0 to the final year for any numerical answer.
• Draw a year-wise table with rows for outlay, sales, costs, depreciation, tax, OCF, working capital and salvage. Step marks are given for each row.
• Write the assumption you make, such as 'working capital fully recovered at end' or 'loss on sale can be set off'. A stated assumption protects marks.
• In MCQs, check whether the answer needs the initial outlay, the yearly OCF or the terminal cash flow. The wrong option is often the correct number for another part.